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The most important open problem in monotone operator theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar’s constraint qualification holds. In this paper, we establish the maximal monotonicity of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}A+B$ provided that $A$ and $B$ are maximally monotone operators such that ${\rm star}({\rm dom}\ A)\cap {\rm int}\, {\rm dom}\, B\neq \varnothing $ , and $A$ is of type (FPV). We show that when also ${\rm dom}\ A$ is convex, the sum operator $A+B$ is also of type (FPV). Our result generalizes and unifies several recent sum theorems.


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[1]Aragón Artacho, F. J., Borwein, J. M., Martín-Márquez, V. and Yao, L., ‘Applications of Convex Analysis within Mathematics’, Math. Program., to appear; doi:10.1007/s10107-013-0707-3.
[2]Attouch, H. and Brezis, H., ‘Duality for the sum of convex functions in general Banach spaces’, in: Aspects of Mathematics and its Applications (ed. Barroso, J. A.) (Elsevier Science Publishers, Amsterdam, 1986), 125–133.
[3]Attouch, H., Riahi, H. and Thera, M., ‘Somme ponctuelle d’operateurs maximaux monotones [Pointwise sum of maximal monotone operators] Well-posedness and stability of variational problems’, Serdica Math. J. 22 (1996), 165190.
[4]Bauschke, H. H. and Combettes, P. L., Convex Analysis and Monotone Operator Theory in Hilbert Spaces (Springer, Berlin, 2011).
[5]Bauschke, H. H., Wang, X. and Yao, L., ‘An answer to S Simons’ question on the maximal monotonicity of the sum of a maximal monotone linear operator and a normal cone operator’, Set-Valued Var. Anal. 17 (2009), 195201.
[6]Bauschke, H. H., Wang, X. and Yao, L., ‘On the maximal monotonicity of the sum of a maximal monotone linear relation and the subdifferential operator of a sublinear function’, in: Proceedings of the Haifa Workshop on Optimization Theory and Related Topics, Contemporary Mathematics, 568 (American Mathematical Society, Providence, RI, 2012), 1926.
[7]Borwein, J. M., ‘Maximal monotonicity via convex analysis’, J. Convex Anal. 13 (2006), 561586.
[8]Borwein, J. M., ‘Maximality of sums of two maximal monotone operators in general Banach space’, Proc. Amer. Math. Soc. 135 (2007), 39173924.
[9]Borwein, J. M., ‘Fifty years of maximal monotonicity’, Optim. Lett. 4 (2010), 473490.
[10]Borwein, J. M. and Lewis, A. S., Convex Analysis and Nonsmooth Optimization, Second expanded edition (Springer, New York, 2005).
[11]Borwein, J. M. and Vanderwerff, J., Convex Functions: Constructions, Characterizations and Counterexamples, Encyclopedia of Mathematics and its Applications vol. 109 (Cambridge University Press, Cambridge, 2010).
[12]Borwein, J. M. and Yao, L., ‘Structure theory for maximally monotone operators with points of continuity’, J. Optim. Theory Appl. 156 (2013), 124 (Invited paper).
[13]Borwein, J. M. and Yao, L., ‘Maximality of the sum of a maximally monotone linear relation and a maximally monotone operator’, Set-Valued Var. Anal. 21 (2013), 603616.
[14]Borwein, J. M. and Yao, L., ‘Some results on the convexity of the closure of the domain of a maximally monotone operator’, Optim. Lett. 8 (2014), 237246.
[15]Borwein, J. M. and Yao, L., ‘Recent progress on Monotone Operator Theory’ Infinite Products of Operators and Their Applications, Contemporary Mathematics, to appear;
[16]Brøndsted, A. and Rockafellar, R. T., ‘On the subdifferentiability of convex functions’, Proc. Amer. Math. Soc. 16 (1995), 605611.
[17]Burachik, R. S. and Iusem, A. N., Set-Valued Mappings and Enlargements of Monotone Operators (Springer, Berlin, 2008).
[18]Butnariu, D. and Iusem, A. N., Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization (Kluwer Academic Publishers, Dordrecht, 2000).
[19]De Bernardi, C. and Veselý, L., ‘On support points and support functionals of convex sets’, Israel J. Math. 171 (2009), 1527.
[20]Fitzpatrick, S., ‘Representing monotone operators by convex functions’, in: Workshop/Miniconference on Functional Analysis and Optimization (Canberra 1988), Proceedings of the Centre for Mathematical Analysis vol. 20 (Australian National University, Canberra, Australia, 1988), 5965.
[21]Holmes, R. B., Geometric Functional Analysis and its Applications (Springer, New York, 1975).
[22]Marques Alves, M. and Svaiter, B.F., ‘A new qualification condition for the maximality of the sum of maximal monotone operators in general Banach spaces’, J. Convex Anal. 19 (2012), 575589.
[23]Phelps, R. R., Convex Functions, Monotone Operators and Differentiability, 2nd edn (Springer, Berlin, 1993).
[24]Phelps, R. R. and Simons, S., ‘Unbounded linear monotone operators on nonreflexive Banach spaces’, J. Nonlinear Convex Anal. 5 (1998), 303328.
[25]Rockafellar, R. T., ‘Local boundedness of nonlinear, monotone operators’, Michigan Math. J. 16 (1969), 397407.
[26]Rockafellar, R. T., Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
[27]Rockafellar, R.T., ‘On the maximality of sums of nonlinear monotone operators’, Trans. Amer. Math. Soc. 149 (1970), 7588.
[28]Rockafellar, R. T. and Wets, R. J.-B., Variational Analysis, 3rd printing (Springer, Berlin, 2009).
[29]Rudin, R., Functional Analysis, 2nd edn (McGraw-Hill, New York, 1991).
[30]Simons, S., Minimax and Monotonicity (Springer, Berlin, 1998).
[31]Simons, S., From Hahn-Banach to Monotonicity (Springer, Berlin, 2008).
[32]Simons, S. and Zǎlinescu, C., ‘Fenchel duality, Fitzpatrick functions and maximal monotonicity’, J. Nonlinear Convex Anal. 6 (2005), 122.
[33]Verona, A. and Verona, M.E., ‘Regular maximal monotone operators’, Set-Valued Anal. 6 (1998), 303312.
[34]Verona, A. and Verona, M.E., ‘Regular maximal monotone operators and the sum theorem’, J. Convex Anal. 7 (2000), 115128.
[35]Verona, A. and Verona, M.E., On the regularity of maximal monotone operators and related results, December 2012.
[36]Veselý, L., ‘A parametric smooth variational principle and support properties of convex sets and functions’, J. Math. Anal. Appl. 350 (2009), 550561.
[37]Voisei, M. D., ‘The sum and chain rules for maximal monotone operators’, Set-Valued Var. Anal. 16 (2008), 461476.
[38]Voisei, M. D., ‘A Sum Theorem for (FPV) operators and normal cones’, J. Math. Anal. Appl. 371 (2010), 661664.
[39]Voisei, M. D. and Zălinescu, C., ‘Strongly-representable monotone operators’, J. Convex Anal. 16 (2009), 10111033.
[40]Voisei, M. D. and Zălinescu, C., ‘Maximal monotonicity criteria for the composition and the sum under weak interiority conditions’, Math. Program. 123 (2010), 265283.
[41]Yao, L., ‘The sum of a maximal monotone operator of type (FPV) and a maximal monotone operator with full domain is maximally monotone’, Nonlinear Anal. 74 (2011), 61446152.
[42]Yao, L., On Monotone Linear Relations and the Sum Problem in Banach Spaces, PhD Thesis, University of British Columbia (Okanagan), 2011,
[43]Yao, L., ‘The sum of a maximally monotone linear relation and the subdifferential of a proper lower semicontinuous convex function is maximally monotone’, Set-Valued Var. Anal. 20 (2012), 155167.
[44]Zălinescu, C., Convex Analysis in General Vector Spaces (World Scientific Publishing, Singapore, 2002).
[45]Zeidler, E., Nonlinear Functional Analysis and its Applications II/A: Linear Monotone Operators (Springer, Berlin, 1990).
[46]Zeidler, E., Nonlinear Functional Analysis and its Applications II/B: Nonlinear Monotone Operators (Springer, Berlin, 1990).
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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